696 BELL SYSTEM TECHNICAL JOURNAL 



with the conditions of constancy of total energy, total number of 

 particles and total number of compartments, is then the product of 

 all the quantities Ws*'- call it W*: 



W* = UWs*. (42) 



s 



As before, and with the same end in view, we form the expression for 

 log W*, and employ Stirling's formula (assuming thus in effect that 

 none of the quantities Zis is smaller than ten or so): 



log W* =^Z log Ws* = ZLQs log Qs - ZZis log ZiJ. (43) 



s s i 



We now take the very great step of identifying not log W of equation 

 (10), but log W*, multiplied by a constant k, with the entropy of the 

 assemblage. 



S = k\og W*. (44) 



Then, when the numbers Zis are changed by small amounts 8Zis, 

 the ensuing change 8E in the total energy E of the assemblage must 

 be linked to the ensuing change in (k log W*) by the equation: 



8S = 8(k log W*) = 8ElT. (45) 



The first variation of {k log W*) is given thus: 



8{k log W*) = - /^I:E(1 + log Zi,)8Zu. (45a) 



5 i 



Let us try the distribution: 



Zi, = a,e-''«/^-=^. (46) 



Substituting this expression into (45a), we get: 



8{k log W*) = - y^EZ(l + log as - ies/kT)8Zis 



s i 



= - ^L(l + log as)i:8Zis + ^Ze.E*5Zi. (47) 



s » -i s i 



= SE/T, 



for the summation ^i8Zis vanishes because the number of compart- 

 ments in each shell is invariable, while the quantity esJ^iZis is equal 

 to the total energy of the particles in the shell s. 



The new statistics, in proposing a new conception of entropy as 

 embodied in equation (44), therefore leads to a new distribution for 

 thermal equilibrium. This distribution is expressed by equation (46) 

 in the new-fashioned way, by stating the number of compartments 



